Seismic imaging involves the estimation of wave propagation velocities in the subsurface from seismic data recorded at the surface. Seismic velocities are related to other physical parameters (for example, density and compressibility, which characterize the lithology of the Earth), and rock-mechanics parameters (for example, porosity and fluid overpressure, which are crucial in reservoir engineering) (Carcione, 2007).
Seismic imaging includes the estimation of both the position of the structures that generate the data recorded at the surface and of a model that describes wave propagation in the subsurface. The two problems are closely related since a model is necessary to infer the position of the reflectors. Waves recorded at the surface are extrapolated in a model of the subsurface (by solving a wave equation) and cross-correlated with a synthetic source wavefield simulated in the same model (Claerbout, 1985). Under the single scattering approximation, reflectors are located where the source and receiver wavefields match in time and space.
Wave-equation tomography (Tarantola, 1984; Woodward, 1992; Biondi and Sava, 1999) is a family of techniques that estimate the velocity model parameters from finite bandwidth signals recorded at the surface. The inversion is usually formulated as an optimization problem where the correct velocity model minimizes an objective function that measures the inconsistency between a trial model and the observations. The objective function can be defined either in the data-space (full-waveform inversion) or in the image-space (migration velocity analysis).
Full-waveform inversion (FWI) (Tarantola, 1984; Pratt, 1999; Sirgue and Pratt, 2004) addresses the estimation problem in the data-space and measures the mismatch between the observations and simulated data. Full-waveform inversion aims to reconstruct the exact model that generates the recorded data. By matching both traveltimes and amplitudes, full-waveform inversion allows one to achieve high-resolution (Sirgue et al., 2010). Nonetheless, a source estimate is needed, the physics of wave propagation (for example, isotropic vs. anisotropic, acoustic vs. elastic, etc.) must be correctly modeled, and a good parameterization (for example, impedance vs. velocity contrasts) is crucial (Kelly et al., 2010). Moreover, an accurate initial model is key to avoid cycle skipping and converge to the global minimum (instead of a local minimum) of the objective function.
Because of the nonlinearity of the wavefields with respect to the velocity model, the objective function in the data domain is highly multimodal (Santosa and Symes, 1989), and local optimization methods can easily get trapped into local minima and fail to converge to the correct model. This is particularly true for reflection full-waveform inversion. Refraction full-waveform inversion focuses on diving waves and mutes the data, retaining only the diving energy (Pratt, 1999). This leads to a better-behaved objective function but requires very long offsets in order to record the refracted energy. Moreover, this approach limits the depth at which a robust inversion result can be expected.
Migration velocity analysis (MVA) (Fowler, 1985; Faye and Jeannot, 1986; Al-Yahya, 1989; Chavent and Jacewitz, 1995; Biondi and Sava, 1999; Sava et al., 2005; Albertin et al., 2006) defines the objective function in the image space and is based on the semblance principle (Al-Yahya, 1989). If the velocity model is correct, images from different experiments must be consistent with each other because a single Earth model generates the recorded data. A measure of consistency is usually computed through conventional semblance (Taner and Koehler, 1969) or differential semblance (Symes and Carazzone, 1991). These two functionals analyze a set of migrated images at fixed locations in space; they consider all the shots that illuminate the points under investigation. Migration velocity analysis leads to smooth objective functions and well-behaved optimization problems (Symes, 1991; Symes and Carazzone, 1991) and is less sensitive than full-waveform inversion to the initial model. On the other hand, because an exemplary embodiment does not use amplitudes in the imaging step, the estimated model has lower resolution than the ideal full-waveform inversion result (Uwe Albertin, personal communication).
Migration velocity analysis measures either the invariance of the migrated images in an auxiliary dimension (reflection angle, shot, etc.) (Al-Yahya, 1989; Rickett and Sava, 2002; Sava and Fomel, 2003; Xie and Yang, 2008) or focusing in an extended space (Rickett and Sava, 2002; Symes, 2008; Sava and Vasconcelos, 2009; Yang and Sava, 2011). All these approaches require the migration of the entire survey in order to analyze the moveout curve in common-image gathers or measure focusing at a specific spatial location. The dimensionality of the (extended) image space and computational complexity of the velocity analysis step rapidly explodes for realistic case scenarios. Moreover, because of the high memory requirement for storing the partial information from each experiment, only a subset of the image points can be considered in the evaluation of the objective function. Illumination holes and/or irregular acquisition geometries can also impact the quality of the common-image gathers but no systematic study of this problem is reported in the literature to our knowledge.
Reverse-time migration (Baysal et al., 1983; McMechan, 1983) is routinely used in exploration geophysics because of its ability to correctly handle the full complexity of the wave propagation phenomena (under the assumption that the physics of wave propagation in the subsurface is correctly modeled). Its computational cost is nevertheless still prohibitive and limits its integration into a migration velocity analysis loop that requires the extraction of common-image gathers for moveout analysis. A migration velocity analysis procedure based on a full-wave propagation engine allows exploitation of more complex wave phenomena (e.g. overturning reflection, prismatic waves, and multiples) and increases the amount of information in the data that can be used (Farmer et al., 2006). Nonetheless, because of the intrinsic cost of wave extrapolation in the time domain, a different approach must be considered for an effective and efficient implementation of velocity model building.
Up to now, the powerful tool offered by reverse-time migration has been used only for reconstructing an image of the impedance contrasts in the subsurface. The velocity model used for generating a migrated image is obtained with other techniques, for example through traveltime tomography (Bishop et al., 1985) or through wave-equation migration velocity analysis (Biondi and Sava, 1999; Sava and Biondi, 2004; Shen and Symes, 2008), which are based on asymptotic approximations and/or linearization of the wave equation operator. These techniques do not exploit all the information encoded in the recorded wavefields because they are either inaccurate in complex velocity models (traveltime tomography) or unable to properly model part of the propagation phenomena (one-way methods). Designing a velocity model-building procedure based on a two-way wave propagation engine will allow an exemplary embodiment to exploit the full complexity of the wavefields and will make the velocity analysis step conceptually consistent with the imaging algorithm used.
Because reverse-time migration is so demanding from a computational and storage point of view, an exemplary embodiment proposes to analyze the semblance of small groups of migrated images from nearby experiments, i.e. neighboring shots or plane-waves with similar ray-parameter or take-off angle, at every image point illuminated by the experiments. Reverse-time migration works only in common-shot and common-receiver configurations, so the choice of the shot-domain for performing migration velocity analysis with a two-way engine may be preferable. Here, we use the word “shot” in its broader sense to include synthetic shot-gathers like plane-wave sources (Whitmore, 1995; Liu et al., 2006; Stoffa et al., 2006; Zhang et al., 2005), random shot-encoded sources (Morton and Ober, 1998; Romero et al., 2000) or any other phase-/amplitude-encoded source (Soubaras, 2006; Perrone and Sava, 2011) in the range of shot-profile migration. Other techniques for velocity model building in the shot-domain use different strategies. Nonetheless, all of them consider the entire survey and migrate all the data before starting the migration velocity analysis loop (deVries and Berkhout, 1984; Yilmaz and Chambers, 1984; Al-Yahya, 1989; Chavent and Jacewitz, 1995; Sava and Biondi, 2004; Shen and Symes, 2008; Symes, 2008; Xie and Yang, 2008). One exemplary embodiment discloses an objective function that evaluates the degree of semblance between images through local correlations in the image space and does not need common-image gathers (CIGs). A morphologic relationship between images is used from nearby experiments to define an objective function that measures shifts in the image space.
The methodology described in this work follows from the linearized wave-equation MVA operator proposed by (Perrone et al., 2012) but removes the linearization of the wave-operator with respect to the model parameters and thus accounts for the full two-way nature of the wavefields. An exemplary embodiment computes the gradient of the objective function with the adjoint-state method (Plessix, 2006). One exemplary advantage of the technique disclosed herein is that it reduces the memory requirements and avoids the need to pick moveout on gathers, and allows the inclusion of all points illuminated by the seismic experiments in the velocity model building.